\section{\label{sec:intro} Introduction}
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Muon dipole moment measurements~\cite{Charpak:1962zz,Bailey:1978mn,Bennett:2006fi,Bennett:2008dy} provide an important test of the Standard Model (SM), where the Brookhaven National Laboratory (BNL) Muon \gmtwo\ Experiment is the last completed experiment in a series of muon anomaly ($a_{\mu}$) and electric dipole moment ($d_{\mu}$) measurements. The Fermi National Accelerator Laboratory (Fermilab) Muon \gmtwo\ Experiment~\cite{Grange:2015fou} is currently collecting and analyzing data, where it has completed the first physics data taking run (\runone) and is now in the second physics data taking run (\runtwo).

The Fermilab experiment can calculate $a_{\mu}$ from~\cite{Grange:2015fou}
\begin{equation}
	a_{\mu}=\left.\left(\frac{g_{e}}{2}\frac{m_{\mu}}{m_{e}}\frac{\omega_{a}}{\left<\omega_{p}\right>}\right)\middle/\left(\frac{\mu_{e}}{\mu_{p}}\right)\right.,
	\label{eq:a_mu_formula}
\end{equation}
where $g_{e}$ is the electron g-factor, $m_{\mu}/m_{e}$ is the muon-electron mass ratio, $\omega_{a}$ is the anomalous precession frequency, and $\mu_{e}/\mu_{p}$ is the electron-proton magnetic moment ratio. The $\left<\omega_{p}\right>$ in Eq.~(\ref{eq:a_mu_formula}) corresponds to the average magnetic field seen by the muons, where the field has been written in terms of the corresponding free proton Larmor frequency because pulsed nuclear magnetic resonance (NMR) is used to measure the field. Values for $g_{e}$, $m_{\mu}/m_{e}$, and $\mu_{e}/\mu_{p}$ can be obtained from CODATA~\cite{Mohr:2015ccw}, where these quantities have relative uncertainties of 0.000\,26, 22, and \SI{3.0}{ppb} respectively. The Fermilab experiment will measure $\omega_{a}$ and $\left<\omega_{p}\right>$ to calculate the corresponding $a_{\mu}$.

The final BNL $a_{\mu}$ value~\cite{Tanabashi:2018oca} has a relative statistical and systematic precision of 460 and 280 parts-per-billion (\SI{}{ppb}) receptively, for a total precision of \SI{540}{ppb} when adding the two uncertainties in quadrature. There is a greater than \SI{3}{\sigma} difference between the BNL measurement and SM prediction~\cite{Tanabashi:2018oca}, which hints at the possibility of physics beyond the SM. The Fermilab experiment has the goal of measuring $a_{\mu}$ with a final statistical and systematic precision of \SI{100}{ppb} respectively, for a total precision of \SI{140}{ppb} when adding the two uncertainties in quadrature. The \SI{100}{ppb} statistical uncertainty is for $\omega_{a}$, as the statistical uncertainty for $\left<\omega_{p}\right>$ is negligible, and the $\omega_{a}$ and $\left<\omega_{p}\right>$ measurements are to each have a \SI{70}{ppb} systematic precision. The BNL $\left|d_{\mu}\right|<\SI{1.8e-19}{\elementarycharge\centi\meter}$~(\SI{95}{\percent} C.L.) measurement provides a factor of 5 improvement in sensitivity with respect to the previous limit, and the Fermilab experiment has the goal improving the sensitivity by at least a 100 times with respect to the BNL measurement.

Achieving a \SI{70}{ppb} systematic precision for $\omega_{a}$ requires a detailed understanding of the muon storage ring beam and spin dynamics. The Fermilab experiment uses the refurbished and upgraded BNL muon storage ring system, where the following discussion describes the storage ring magnetic and electric guide fields, lattice functions, tunes, closed orbits, polarized muon injection, and storage of the muons. The scintillating-fiber beam monitor (FBM), straw tracker, and calorimeter systems are also discussed, as these systems provide information about the storage ring beam. Finally, an overview of the known $\omega_{a}$ systematic effects that arise from beam and spin dynamics is also given.
